In this section we fix our notation and collect basic facts from convex geometry.
General references for the theory of convex bodies are the books by Gardner [1]}, Gruber [2]}, Schneider [3]}. We write \(\mathcal {K}^n\)  for the set of convex bodies (compact convex subsets) of \(\mathbb {R}^n\) . We write \(\mathcal {K}^n_o\)  for the set of convex bodies that contain the origin in their interiors. A compact set \(K\subset \mathbb {R}^n\)  is a star-shaped set (with respect to the origin) if the intersection of every straight line through the origin with \(K\)  is a line segment. Let \(K\subset \mathbb {R}^n\)  be a compact star shaped set (with respect to the origin), the radial function \(\rho (K,\cdot ):\mathbb {R}^n\backslash \lbrace 0\rbrace \rightarrow \mathbb {R}\)  is defined by
\(\rho (K,x)=\rho _K(x)=\max \lbrace \lambda \ge 0:\; \lambda x\in K\rbrace .\) 
